The spectral edge of sparse directed Erd\"os-R\'enyi graphs

Let $d>1$ be fixed and let $A_n$ be an $n\times n$ matrix with independent $\Ber(d/n)$ entries. For every $0<r<\sqrt d$, we prove that, with high probability, a positive proportion of the eigenvalues of $A_n$ have modulus larger than $r$. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to $\sqrt d$. Our proof works with the Brown measure $\mu_d$ of the adjacency operator of the directed Poisson--Galton--Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of $A_n$ to $\mu_d$ in probability. We prove that the outer radius of its support is $\sqrt d$, using a resolvent recursion on Poisson--Galton--Watson trees.